Ayn Rand and Leonard Peikoff have answered that question several times. No individual, by obligation, is responsible for any other human being unless they're your children. Now then, what would happen to the mentally challenged people who have no loved one to take care of them? They'll depend on voluntary charity. Just because it is a free society, it does not mean that no one will be charitable. It would be bad if charity was an obligation but voluntarily it's ok. Now just because its not an obligation it wouldn't mean that nobody will do it. Along history it has been proved that voluntary charity exist even when government does it and in most cases private charity works more efficiently than the other.

I'm sure that in a free objectivist society there would be some people helping the ones in need because of mental disfunctions but they wouldn't be altruistic. There would be some kind of cost and profit of it somehow. But in short the answer is what I said at the beginning: nobody is responsible for the mentally challenged unless someone wants to do it. One may say that it isn't fair because they were born that way, it is also not fair to place the burden of their problem to other people who have done nothing wrong to them. That, in fact, would be slavery.

a) Even if those cases become real there's nothing to do. It may be sad but no individual has the obligation to help any other one. That's that.

b) Those cases won't occur: this is what I meant when saying that in a objectivist society helping someone in need won't be altruistic. Objectivists businessman would see there is a demand for some kind of facility to take care of such people and they'll find a way to satisfy the demand and profit from it.

I asked a question that can be linked to this one that didn't really get a great answer. I asked if there was any statistical data that proved when the economy does well, the amount of charitable donations increases.

I can't really know for sure about that but it sounds reasonable. Also something that might interest you, I once read about how people are less encouraged to give charitable donations when the government provides health care, good or bad. This comes from the notion that if there is someone taking the responsibility by using the money of our taxes, then we don't have that responsibility at all. Of course, if people see that nobody is doing anything and people are doing well, economically, they may be encouraged to help since they can afford it without meaningful sacrifice.

And of course, in some way it would benefit society as a whole, even yourself and that's what you want. Consider the possibility of businessmen offering price reductions or discounts on products or services for donors to X organization.